Abstract. In this paper we present a numerical method to compute Diophantine
rotation numbers of circle maps with high accuracy. We mainly
focus on analytic circle diffeomorphisms, but the method also
works in the case of (enough) finite differentiability. The
keystone of the method is that, under these conditions, the map is
conjugate to a rigid rotation of the circle. Moreover, albeit it
is not fully justified by our construction, the method turns to
be quite efficient for computing rational rotation numbers. We
discuss the method through several numerical examples.